Information Technology Reference
In-Depth Information
•
(
,
)
∈
(
,
)
∈
⃒
(
,
)
∈
If
R
is transitive,
x
y
R
&
y
z
R
x
z
R
, is reflected by
μ
R
(
,
)
=
μ
R
(
,
)
=
⃒
μ
R
(
,
)
=
x
y
y
z
1
x
z
1, that implies,
T
(μ
R
(
x
,
y
), μ
R
(
y
,
z
))
μ
R
(
x
,
z
),
for all t-norms
T
. Notice that if
μ
R
(
x
,
y
)
=
0or
μ
R
(
y
,
z
)
=
0, then, for example,
T
(μ
R
(
x
,
y
), μ
R
(
y
,
z
))
=
T
(
0
, μ
R
(
y
,
z
))
=
0
μ
R
(
x
,
z
).
Hence, for all
x
,
y
,
z
in
X
, and any continuous t-norm
T
,is
T
(μ
R
(
x
,
y
), μ
R
(
y
,
z
))
μ
R
(
x
,
z
).
that reflects equationally the transitivity of
R
.
Example 4.3.2
1. The matrix
⊛
⊞
11
/
82
/
8
⊝
⊠
[
μ
]=
1
/
813
/
8
2
/
83
/
81
is reflexive and symmetric (
fuzzy similarity
). In addition,
[
μ
]↗
pr od
[
μ
]=[
μ
]
,
that is,
[
μ
]
is prod-transitive. Notice that,
⊛
⊞
11
/
82
/
8
⊝
⊠
=[
μ
]
,
[
μ
]↗
min
[
μ
]=
1
/
813
/
8
2
/
82
/
81
and
is also
W
-
transitive. Notice that this last matrix is reflexive, non symmetric, but min-
transitive, since
⊛
⊝
[
μ
]
is not min-transitive. Of course, since
W
pr od
,
[
μ
]
⊞
⊠
↗
min
⊛
⊝
⊞
⊠
=
⊛
⊝
⊞
⊠
,
/
/
/
/
/
/
11
82
8
11
82
8
12
82
8
/
/
/
/
/
/
1
813
8
1
813
8
2
813
8
2
/
82
/
81
2
/
82
/
81
2
/
82
/
81
3-matrix reflects a min-
preorder
and, consequently, a
T
-preorder
for all t-norm
T
.
2. The before mentioned fuzzy relation
hence, this 3
×
is not only
reflexive and symmetric, but also
W
-transitive, as it can be proved by distinguish-
ing the four cases:
k
μ(
x
,
y
)
=
max
(
0
,
1
−
k
|
x
−
y
|
)
,
k
;
k
<
|
,
k
<
|
;
k
≥|
x
−
y
|
≥|
y
−
z
|
x
−
y
|
y
−
z
|
≥|
x
−
y
|
,
1
; and
k
<
|
1
k
k
<
|
y
−
z
|
x
−
y
|
,
≥|
y
−
z
|
.
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